Exercise 4.12Z: White Gaussian Noise

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Power spectral densities
of white noise

A noise signal $n(t)$  is called white; if it contains all spectral components without preference of any frequencies.

• The physical Power spectral density defined only for positive frequencies $f$  ${\it \Phi}_{n+}(f)$  is constant  $($equal  $N_0)$  and extends frequency-wise to infinity.
• ${\it \Phi}_{n+}(f)$  is shown in green in the upper graph.  The plus sign in the index is to indicate that the function is valid only for positive values of $f$  .
• For mathematical description one usually uses the two-sided Power spectral density spectrum  ${\it \Phi}_{n}(f)$.  Here applies for;all frequencies from  $-\infty$  to  $+\infty$  (blue curve in the upper picture):

$${\it \Phi}_n (f) ={N_0}/{2}.$$

The bottom graph shows the two Power spectral densities  ${\it \Phi}_{b}(f)$  and  ${\it \Phi}_{b+}(f)$  of a bandlimited white noise signal  $b(t)$  It holds with the one-sided bandwidth  $B$:

$${\it \Phi}_b(f)=\left\{ {N_0/2\atop 0}{\hspace{0.5cm} {\rm f\ddot{u}r}\quad |f|\le B \atop {\rm else}}\right.,$$

$${\it \Phi}_{b+}(f)=\left\{ {N_0\atop 0}{\hspace{0.5cm} {\rm f\ddot{u}r}\quad 0 \le f\le B \atop {\rm else}}\right.$$

For computer simulation of noise processes, band-limited noise must always be assumed, since only discrete-time processes can be handled. For this, the  SamplingTheorem  must be obeyed.  This states that the bandwidth  $B$  must be set according to the pitch  $T_{\rm A}$  of the simulation.

Assume the following numerical values throughout the exercise:

• The noise power density –  with respect to the resistor  $1 \hspace{0.05cm}\rm \Omega$  –  beträgt  $N_0 = 4 \cdot 10^{-14}\hspace{0.05cm}\rm V^2/Hz$.
• The (one-sided) bandwidth of the band-limited white noise is $B = 100 \hspace{0.08cm}\rm MHz$.

Hints:

• This exercise belongs to the chapter  Power-Spectral Density.
• Reference is also made to the chapter  Auto-correlation function.
• The properties of white noise are summarized in the second part of the tutorial video  The AWGN channel (German) .

Questions

1

Which statements are always true for a white noise signal  $n(t)$  Give reasons for your answers.

	The ACF  $\varphi_n(t)$  has a si-shaped progression.
	The ACF  $\varphi_n(\tau)$  is a Dirac at  $\tau = 0$  with weight  $N_0/2$.
	In practice, there is no (exact) white noise.
	Thermal noise can always be approximated as white.
	White noise is always Gaussian distributed.

2

Calculate the ACF  $\varphi_b(\tau)$  of the random signal $b(t)$ bandlimited to  $B = 100 \hspace{0.08cm}\rm MHz$  What value results for  $\tau = 0$?

$\varphi_b(\tau = 0) \ = \ $ $\ \cdot 10^{-6} \ \rm V^2$

3

What is the rms value of this bandlimited random signal  $b(t)$?

$\sigma_b \ = \ $ $\ \rm mV$

4

What sampling distance  $T_{\rm A}$  should be chosen (at most) if the band-limited signal  $b(t)$  is used for discrete-time simulation of white noise?

$T_{\rm A} \ = \ $ $\ \rm ns$

5

Assume sampling distance  $T_{\rm A} = 1 \hspace{0.05cm}\rm ns$  Then, which of the statements are true for two consecutive samples of the signal  $b(t)$ ?

	The samples are uncorrelated.
	The samples are positively correlated.
	The samples are negatively correlated.

Solution

Solution

(1)  Correct are solutions 2, 3, and 4:

• The auto-correlation function (ACF) is the Fourier transform of the Power spectral density (PSD). Here:

$${\it \Phi}_n (f) = {N_0}/{2} \hspace{0.3cm} \bullet\!\!-\!\!\!-\!\!\!-\!\!\circ\, \hspace{0.3cm} \varphi_n (\tau)={N_0}/{2} \cdot {\rm \delta} ( \tau).$$

• However, there is no "real" white noise in physics, since such a noise would have to have an infinitely large signal power  $($the integral over the PSD as well as the ACF value at  $\tau = 0$  are both infinitely large$)$.
• Thermal noise has a constant PSD up to frequencies of about  $\text{6000 GHz}$ . Since all (current) üb transmission systems operate in a much lower frequency range, thermal noise can be said to be "white" to a good approximation.
• The statistical property "white" says nothing about the amplitude distribution, which is determined by the probability density function (PDF) alone.
• When considering the phase of a bandpass signal as the stochastic variable, it is often modeled as uniformly distributed between  $0$  and  $2\pi$  .
• If there are no statistical bindings between the respective phase angles at different times, this random process is also "white".

ACF of band-limited noise

(2)  The Power spectral density spectrum is a rectangle of width  $2B$  and height  $N_0/2$.

• The inverse Fourier transformation yields an si function:

$$\varphi_b(\tau) = N_0 \cdot B \cdot {\rm si} (2 \pi B \tau)\hspace{0.3cm} \Rightarrow \hspace{0.3cm}\varphi_b(\tau = 0) = N_0 \cdot B \hspace{0.15cm}\underline {=4}\cdot 10^{-6} \ \rm V^2.$$

(3)  The ACF value at the point  $\tau = 0$  gives the power. 

• The root of this is called the rms value:

$$\sigma_b = \sqrt{\varphi_b(\tau = 0)} \hspace{0.15cm}\underline {=2 \hspace{0.05cm}\rm V}.$$

(4)  The ACF computed in  (3)  has zeros at equidistant distance from  $T_{\rm A}= 1/(2B)\hspace{0.15cm}\underline {=5\hspace{0.05cm} \rm ns}$: 

• There are no statistical bindings between the two signal values  $b(t)$  and  $b(t + \nu \cdot T_{\rm A})$,
• where  $\nu$  can take all integer values.

(5)  The correct solution is suggested solution 2.

• The ACF value at  $\tau = T_{\rm A} = 1 \hspace{0.05cm}\rm ns$  amounts to.

$$\varphi_b(\tau = T_{\rm A}) = {\rm 4 \cdot 10^{-6} \hspace{0.1cm}V^2 \cdot si (\pi/5) \approx 3.742 \cdot 10^{-6} \hspace{0.1cm}V^2} > 0.$$

• This result says:   Two signal values separated by  $T_{\rm A} = 1 \hspace{0.05cm}\rm ns$  are positively correlated:
• If  $b(t)$  is positive and large;, then with high probability  $b(t+1 \hspace{0.05cm}\rm ns)$  is also positive and large;.
• In contrast, there is a negative correlation between  $b(t)$  and  $b(t+7 \hspace{0.05cm}\rm ns)$  If  $b(t)$  is positive, then  $b(t+7 \hspace{0.05cm}\rm ns)$  is probably negative.